Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth

Theorem 1.1

Assume that 2 < p < 4 and ( f 5 ) – ( f 8 ) hold, then there exist μ 0 > 0 such that problem ( P ) has a positive and radial ground-state solution for any 0 < μ < μ 0 . This ground-state is a (local) minimizer of E μ in the set V ( a ) . In addition, any ground-state for E μ on S ( a ) is a local minimizer of E μ on V ( a ) . Moreover, there exists κ 0 > 0 such that, for any κ ≥ κ 0 , where κ is given in ( f 7 ) , problem ( P ) has a second solution v ∈ S ( a ) , which is the mountain path-type solution and satisfies E μ ( v ) > 0 .

Theorem 1.2

Let p = 4 and ( f 5 ) – ( f 8 ) hold, then there exists a 1 > 0 , μ 1 > 0 , and κ 1 > 0 such that E μ restricted to S ( a ) has a ground-state solution if 0 < μ < μ 1 and κ > κ 1 , or 0 < a < a 1 and κ > κ 1 , which is the mountain path-type solution.

Theorem 1.3

Assume that 2 < p ≤ 4 and ( f 5 ) – ( f 8 ) hold. Let u μ , a be the positive and radial ground-state solution in Theorem 1.1. Set v μ and w a , μ as the mountain path-type solution in Theorems 1.1  and 1.2, respectively. Then, the following assertions hold:

1. ‖ ∇ u μ , a ‖ 2 → 0 and the corresponding Lagrange multiplier λ μ , a → 0 as μ → 0 + ,

2. u μ , a → 0 , in H 1 ( R 2 ) and λ μ , a → 0 as a → 0 + . Let U be the unique positive solution of

   (1.5) − Δ U + U = μ U p − 1 , in R 2 , lim ∣ x ∣ → ∞ U ( x ) = 0 ,

   and

   U μ , a ( x ) ≔ λ μ , a 1 2 − p u μ , a x λ μ , a ,

   then U μ , a → U in H 1 ( R 2 ) as a → 0 + .

3. v μ → v in H 1 ( R 2 ) as μ → 0 + , where v is a ground-state solution of Problem (1.2) for κ > κ 0 .

4. w a , μ → 0 in H 1 ( R 2 ) as a → 0 + . Moreover, w a , μ → w in H 1 ( R 2 ) as μ → 0 + , where w is also a ground-state solution of Problem (1.2) for κ > κ 1 .

Remark 1.1

To consider the asymptotic behavior of solution in Theorem 1.3(ii), we are inspired by the results of [21, Section 4], where the key points are proving that λ μ , a → 0 as a → 0 + and the uniform boundedness of ∥ u μ , a ∥ ∞ . However, in our case, the nonlinearity is Moser-Trudinger critical growth at infinity and there is no corresponding limit equation at infinity. And thus, we cannot deduce the uniform boundedness of ∥ u μ , a ∥ ∞ as that in [21], where the nonlinearity is subcritical growth at infinity and there is the corresponding limit equation at infinity. We have to take a different way to attack it.

Remark 1.2

As a reference model, take

which satisfies ( f 5 ) – ( f 8 ) . One can check that

and the function t ↦ K ′ ( t ) t − 4 K ( t ) t p is strictly increasing in ( 0 , ∞ ) , where

Lemma 2.1

Let β > 0 and s > 1 . Then, for each α > s , there exists a positive constant C = C ( α ) such that for all t ∈ R ,

In particular, ( e β u 2 − 1 ) s belongs to L 1 ( R 2 ) for any u ∈ H 1 ( R 2 ) .

Proof

See the proof in [13, Lemma 2.2].□

Lemma 2.2

Let u ∈ S ( a ) be arbitrary but fixed, then we have

1. ‖ ∇ u t ‖ 2 → 0 and E μ ( u t ) → 0 , as t → 0 ,

2. ‖ ∇ u t ‖ 2 → + ∞ and E μ ( u t ) → − ∞ , as t → + ∞ .

Proof

By a straightforward calculation, it follows that

and

For any fixed s > 2 , we have

It follows from ( f 5 ) – ( f 6 ) that for any ε > 0 , there exist C ε > 0 and s > 4 such that

By (2.2) and (2.7),

Recall the famous Gagliardo-Nirenberg inequality [34, Lemma 2.4],

where r = N ( 1 2 − 1 p ) . According to (2.4), it follows that

and then

In order to show ( i i ) , note that as t → + ∞ , ‖ ∇ u t ‖ 2 → + ∞ and

where we used the fact that q > 4 .□

Lemma 3.1

For every u ∈ S ( a ) , the function Ψ u has exactly two critical points t u − and t u + with 0 < t u − < t u + . Moreover,

1. t u − is a local minimum point for Ψ u ( t ) = E μ ( u t ) , Ψ u ( u t u − ) < 0 , and u t u − ∈ V ( a ) ;

2. t u + is a global maximum point for Ψ u , Ψ u ′ ( t ) < 0 for all t > t u + and

   E μ ( u t u + ) ≥ inf u ∈ ∂ V ( c ) E μ ( u ) > 0 ,

for a suitable constant ρ > 0 .

Proof

In view of the Gagliardo-Nirenberg inequality, one has

Let 0 < ρ ≤ 1 6 and μ ≤ ρ 4 − p + ε for some ε > 0 . From (2.2), (2.6), and (3.1), one has

for small enough ρ . Because Ψ u ( t ) → 0 − , ∥ ∇ u t ∥ 2 → 0 as t → 0 and Ψ u ( t ) = E μ ( u t ) > 0 when u t ∈ ∂ V ( a ) = { v ∈ S ( a ) : ‖ ∇ v ‖ 2 2 = ρ 2 > 0 } . Necessarily, Ψ u ′ has a first zero t u − > 0 corresponding to a local minima. In particular, u t u − ∈ V ( a ) and Ψ u ( t u − ) < 0 . Now, from Ψ u ( t u − ) < 0 , Ψ u ( t ) > 0 when u t ∈ ∂ V ( a ) and Ψ u ( t ) → − ∞ as t → ∞ , Ψ u ′ has a second zero t u + > t u − corresponding to a local maxima of Ψ u with E μ ( u t u + ) ≥ inf u ∈ ∂ V ( a ) E μ ( u ) > 0 . To conclude the proofs of (i) and (ii), it just suffices to show that Ψ u ′ has at most two zeros. However, this is equivalent to show that

has at most two zeros. Note that

and there exists t * > 0 such that θ ( t * ) > 0 from (3.2). Since ( f 8 ) ,

has a unique zero point, which implies that θ ( t ) has indeed at most two zero points.□

Proposition 3.1

Assume that 2 < p < 4 and ( f 5 ) – ( f 8 ) hold. Then, E μ restricted to S ( a ) has a positive and radial ground-state for any μ < μ 0 * . This ground-state is a (local) minimizer of E μ in the set V ( a ) . In addition, any ground-state for E μ on S ( a ) is a local minimizer of E μ on V ( a ) .

Proof

From the definition of m μ ( a ) , there exists a minimizing sequence { u n } ⊂ V ( a ) such that E μ ( u n ) → m μ ( a ) as n → ∞ . For any fixed n , denoting by v n the Schwarz rearrangement of ∣ u n ∣ . For more details, we refer readers to [6,26]. It follows from Lemma 3.1 and the property of v n that v n ∈ S ( a ) ∩ V ( a ) and { v n } is also the minimizing sequence. Note that, { v n } is bounded in H 1 ( R 2 ) . Indeed, there exists v ∈ V ( a ) such that as n → ∞ , v n ⇀ v in H 1 ( R 2 ) and v n → v in L q ( R 2 ) , q ∈ ( 2 , + ∞ ) . By (2.2), we have